Question

apply the properties of integer exponent to generate equivalent expressions to (7^3)^7 times 7^20 divided by 6^-2 with only positive exponents, then solve the expression. provide the answer as an improper fraction

Answer 1

To apply the properties of integer exponents, we need to simplify each term individually and then combine them.
Let's break down each term and simplify:

First, simplify (7^3)^7 as 7^(3*7) = 7^21.
Next, simplify 7^20 as (7^2)^10 = 49^10.
Lastly, simplify 6^-2 as 1/(6^2) = 1/36.

Now, combine the simplified terms:
(7^21)*(49^10)/(1/36)

To multiply the terms, add the exponents of like bases:
7^21 * (7^2)^10 * 49^10 / (1/36)
= 7^21 * 7^20 * 49^10 / (1/36)
= 7^(21+20) * 49^10 / (1/36)
= 7^41 * 49^10 / (1/36).

To divide, subtract the exponents of like bases:
7^41 * 49^10 / (1/36)
= (7^41 * 49^10) * 36.

Now, let's evaluate this expression:
7^41 * 49^10 * 36
= (2^3)^41 * (7^2)^10 * 36
= 2^(3*41) * 7^(2*10) * 36
= 2^123 * 7^20 * 36.

Therefore, the equivalent expression with only positive exponents is 2^123 * 7^20 * 36.